PARIPolynomial should be converted back to GAP polynomial

[videlec]

For now, conversion to pari polynomials are converted into vectors

gap> LoadPackage("PARIInterface");;
gap> x := Indeterminate(Rationals, "x");;
gap> p := PARIPolynomial(3*x^2 + 2*x + 1);
PARI(3*x^2 + 2*x + 1)
gap> PARI_GEN_TO_OBJ(p);
[ 1, 2, 3 ]

It would be nicer if we would get back the original

Also, conversion to pari polynomials forget about variable names

gap> LoadPackage("PARIInterface");
true
gap> PARI_UNIPOLY([1,2,3]);
PARI(3*x^2 + 2*x + 1)
gap> x := Indeterminate(Rationals, "x");
x
gap> y := Indeterminate(Rationals, "y");
y
gap> PARIPolynomial(x + 1);
PARI(x + 1)
gap> PARIPolynomial(y + 1);
PARI(x + 1)

If there is no workaround, that should be properly documented.

[markuspf]

Does pari have a way to set variable names, or does it have any other way of identifying variables?

I think we can make this consistent and more useful.

[fingolfin]

In general, since PARI and GAP support different coefficient domains, I don't think this conversion will always be useful / possible. In other similar situations (interfacing one computer algebra system from another) it turned out that for practical computations, you almost never perform conversions, and those usually only happen at the start (to generate input) and the end, i.e., they don't need to be very efficient. Instead, it's better to keep data in the external system, and process it here.

Specifically:

• add +, *, -, AINV, ... method for PARIPolynomial and other PARI data types
• provide other useful PARI operations that take PARIPolynomials as inputs
• one or more methods for operation Value to evaluate PARI polynomials might be nice
• two functions to convert PARI polynomials to GAP, and vice versa, if possible; presumably these would be given a list of (GAP) indeterminates, and (if there is such a thing) PARI indeterminates, to decide what is mapped how.

All in all, I usually would avoid any "default" conversions, other than in very specific special cases where they really make sense.